论文标题
关于三角类别的刚性子类别的索引
The index with respect to a rigid subcategory of a triangulated category
论文作者
论文摘要
帕鲁（Palu）定义了相对于合适的三角类别中的集群倾斜对象的索引，以便更好地理解caldero-chapoton图，该图表在群集代数和表示理论之间表现出连接。我们通过提出有关违反有限，刚性子类别的索引来进一步推动这一点，并且我们表明该索引的行为与经典索引相似。 令$ \ mathcal {c} $为具有拆分idempotents的骨骼小三角形类别，因此，它是一个外部构造类别$（\ Mathcal {C}，\ Mathbb {e}，\ Mathfrak {s}）$。假设$ \ Mathcal {x} $是$ \ Mathcal {C} $中的违反有限的刚性子类别。我们定义了对象$ c \ in \ Mathcal {c} $的索引$ \ Mathrm {ind} _ {\ Mathcal {x}}（c）$相对于$ \ Mathcal {x} $，作为$ \ Mathcal {x} $作为$ k_ {0} $ [0} $ - 类$ [c] $ [c] $ k_ {0}（\ Mathcal {C}，\ Mathbb {e} _ {\ Mathcal {x}}，\ Mathfrak {s} _ {\ Mathcal {x}} $ $（\ Mathcal {C}，\ Mathbb {E} _ {\ Mathcal {X}}}，\ Mathfrak {s} _ {\ Mathcal {x}}）$。与经典案例相比，我们在$ \ mathcal {c} $中给出了$ \ mathrm {indrm {ind} _ {\ mathcal {x}} $的错误项的添加性公式。 如果$ \ MATHCAL {X} $包含在另一个合适的子类别中$ \ Mathcal {t} $ of $ \ Mathcal {C} $，则有一个过滤$ q \ colon K_ {0}（\ Mathcal {C}，\ Mathbb {E} _ {\ Mathcal {t}}，\ Mathfrak {s} _ {\ Mathcal {t}}}） K_ {0}（\ Mathcal {C}，\ Mathbb {E} _ {\ Mathcal {X}}，\ Mathfrak {s} _ {\ Mathcal {x}}}）$。因此，为了描述$ k_ {0}（\ Mathcal {C}，\ Mathbb {e} _ {\ Mathcal {x}}}，\ Mathfrak {s} _ {\ Mathcal {x Mathcal {x}}） $ k_ {0}（\ Mathcal {C}，\ Mathbb {e} _ {\ Mathcal {t}}，\ Mathfrak {s} _ {\ Mathcal {t}}}} $和$ \ operatornAme {ker} q $。我们在某些假设下这样做。
Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero-Chapoton map that exhibits the connection between cluster algebras and representation theory. We push this further by proposing an index with respect to a contravariantly finite, rigid subcategory, and we show this index behaves similarly to the classical index. Let $\mathcal{C}$ be a skeletally small triangulated category with split idempotents, which is thus an extriangulated category $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Suppose $\mathcal{X}$ is a contravariantly finite, rigid subcategory in $\mathcal{C}$. We define the index $\mathrm{ind}_{\mathcal{X}}(C)$ of an object $C\in\mathcal{C}$ with respect to $\mathcal{X}$ as the $K_{0}$-class $[C]_{\mathcal{X}}$ in Grothendieck group $K_{0}(\mathcal{C},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$ of the relative extriangulated category $(\mathcal{C},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$. By analogy to the classical case, we give an additivity formula with error term for $\mathrm{ind}_{\mathcal{X}}$ on triangles in $\mathcal{C}$. In case $\mathcal{X}$ is contained in another suitable subcategory $\mathcal{T}$ of $\mathcal{C}$, there is a surjection $Q\colon K_{0}(\mathcal{C},\mathbb{E}_{\mathcal{T}},\mathfrak{s}_{\mathcal{T}}) \twoheadrightarrow K_{0}(\mathcal{C},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$. Thus, in order to describe $K_{0}(\mathcal{C},\mathbb{E}_{\mathcal{X}},\mathfrak{s}_{\mathcal{X}})$, it suffices to determine $K_{0}(\mathcal{C},\mathbb{E}_{\mathcal{T}},\mathfrak{s}_{\mathcal{T}})$ and $\operatorname{Ker} Q$. We do this under certain assumptions.
